Question

Factorise the following: x4 + 8x²y² + 16y4​

Answer 1

Answer:

STEP1:Equation at the end of step 1 ((x4)-((8•(x2))•(y2)))+24y4 STEP 2 :Equation at the end of step2: ((x4) - (23x2 • y2)) + 24y4 STEP3:Trying to factor a multi variable polynomial

 3.1    Factoring    x4 - 8x2y2 + 16y4 

Try to factor this multi-variable trinomial using trial and error 

 Found a factorization  :  (x2 - 4y2)•(x2 - 4y2)

Detecting a perfect square :

 3.2    x4  -8x2y2  +16y4  is a perfect square 

 It factors into  (x2-4y2)•(x2-4y2)

which is another way of writing  (x2-4y2)2

How to recognize a perfect square trinomial:  

 • It has three terms  

 • Two of its terms are perfect squares themselves  

 • The remaining term is twice the product of the square roots of the other two terms

Trying to factor as a Difference of Squares:

 3.3      Factoring:  x2-4y2 

Put the exponent aside, try to factor  x2-4y2 

Theory : A difference of two perfect squares,  A2 - B2  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =

         A2 - AB + BA - B2 =

         A2 - AB + AB - B2 =

         A2 - B2

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 4 is the square of 2

Check :  x2  is the square of  x1 

Check :  y2  is the square of  y1 

Factorization is :       (x + 2y)  •  (x - 2y) 

Raise to the exponent which was put aside

Factorization becomes :   (x + 2y)2   •  (x - 2y)2  

Final result :

(x + 2y)2 • (x - 2y)2

Answer 2

$\large\underline{\sf{Solution-}}$

Given algebraic expression is

$\rm \: {x}^{y} + 8 {x}^{2} {y}^{2} + {16y}^{4}$

can be rewritten as

$\rm \: = \: {( {x}^{2} )}^{2} +8{x}^{2} {y}^{2} + {( {4y}^{2} )}^{2}$

$\rm \: = \: {( {x}^{2} )}^{2} +2 \times {x}^{2} \times {4y}^{2} + {( {4y}^{2} )}^{2}$

We know,

$\boxed{ \rm{ \: {(a + b)}^{2} = {a}^{2} + 2ab + {b}^{2} \: }}$

So, here

$\rm \: a = {x}^{2}$

and

$\rm \: b = {4y}^{2}$

So, using this identity, we get

$\rm \: = \: {( {x}^{2} + {4y}^{2})}^{2}$

$\rm \: = \: ( {x}^{2} + {4y}^{2})( {x}^{2} + {4y}^{2})$

Hence,

$\rm\implies\:\boxed{\rm{{x}^{4}+{8x}^{2}{y}^{2}+{16y}^{4}=({x}^{2} + {4y}^{2})( {x}^{2}+{4y}^{2})}}$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}$